Lyapunov Stability - Definition For Continuous-time Systems

Definition For Continuous-time Systems

Consider an autonomous nonlinear dynamical system

,

where denotes the system state vector, an open set containing the origin, and continuous on . Suppose has an equilibrium .

1. The equilibrium of the above system is said to be Lyapunov stable, if, for every, there exists a such that, if, then, for every .
2. The equilibrium of the above system is said to be asymptotically stable if it is Lyapunov stable and if there exists such that if, then .
3. The equilibrium of the above system is said to be exponentially stable if it is asymptotically stable and if there exist such that if, then, for .

Conceptually, the meanings of the above terms are the following:

1. Lyapunov stability of an equilibrium means that solutions starting "close enough" to the equilibrium (within a distance from it) remain "close enough" forever (within a distance from it). Note that this must be true for any that one may want to choose.
2. Asymptotic stability means that solutions that start close enough not only remain close enough but also eventually converge to the equilibrium.
3. Exponential stability means that solutions not only converge, but in fact converge faster than or at least as fast as a particular known rate .

The trajectory x is (locally) attractive if

for for all trajectories that start close enough, and globally attractive if this property holds for all trajectories.

That is, if x belongs to the interior of its stable manifold. It is asymptotically stable if it is both attractive and stable. (There are counterexamples showing that attractivity does not imply asymptotic stability. Such examples are easy to create using homoclinic connections.)